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Crystallography Topology @ Jewel Info 4 U
By:Ritika
Summary: Crystallography refers
to the analysis of atoms in crystals
and topology refers to the study of
distortion and invariant
connectivity characteristics of
mathematical objects. Thus,
crystallographic topology deals with
association of the two attributes
together. The topology of
crystallographic groups is
approached using orbifolds, and that
of simple crystal structures using
Morse functions on orbifolds.
Orbifolds
W. D. Dunbar is considered as the
father of orbifold as he started the
first study of orbifolds in
1981.Crystallographic orbifold O is
defined as the quotient space of
either a sphere SP, or Euclidean EU,
space modulo a discrete
crystallographic symmetry group, G
and is denoted as
O= (SP or EU)/G
An orbifold comprises a basic
topological space along with a
singular set embedded in it.
Moreover, an interesting fact is
that a properly bound fundamental
domain within a space group's unit
cell is an orbifold.
Crystallographic Orbifolds Types
There are three types of groups in
which general crystallography could
be divided in terms of
crystallographic orbifolds. They
are:
* Point Groups: This is used
for explaining the symmetry of
special positions within a space
group and is also termed as Wyckoff
site symmetries.
* Plane Groups: The space
groups projected along their primary
axes of symmetry become plane
groups.
* Space Groups: In case of
space groups, all of them have a
parent point group. The main use of
space groups are for classification
of schemes. It is important to note
that order of a space group is
always infinite.
The crystallographic orbifolds
associated respectively with the
above groups are elliptic
2-orbifolds,Euclidean 2-orbifolds,
and Euclidean 3-orbifolds. Thus, the
above are referred to as point
orbifolds, plane orbifolds, and
space orbifolds respectively.
Critical Nets and Orbifolds
Morse functions are the basis of
critical net in crystallography and
this plays a vital role in crystal
chemistry and crystallographic
topology. Critical net is defined as
mathematical mapping from Euclidean
3-space to Euclidean 1-space. This
could be used to orbifold so that
the Euclidean 1-space of density is
deformed vertical in the page. The
critical-net-on-orbifold model
features the conventional
crystallographic invariant lattice
complexes and permits concise
quotient-space topological figures
to be drawn without any repetitions
that are attributed to normal
crystal structure figures.
Lattice Complex
Lattice Complex takes a vital role
in crystallography and the history
of lattice complex started many
years before in the branch of
crystallography. Lattice complexes
refer to the configurations of
points that recur at least once but
generally repeatedly throughout the
family of all space groups. It is
important to note that points on
symmetry elements have smaller total
unit cell occupancy and this is
called the Wyckoff site
multiplicity.
The critical points are best
described as representing 0-, 1-,
2-, and 3-dimensional cells in a
topological Morse function. In this,
generally non-degenerate crucial
points are taken into consideration
here because a degenerate crucial
point can at all times be distorted
into a series of non-degenerate ones
via the morsification process. A
degenerate critical point will have
a singular second derivative matrix
with one or more zero or nearly zero
eigen values. In fact, the critical
points are present where the first
derivative with respect to global
density will be zero. Also, a 3-3
symmetric matrix occurs as the
second derivative at that particular
point. Moreover, purely when the
critical point is fully
non-degenerate, will this have a
non-zero determinant.
Color Crystallographic Groups
The group or normal-subgroup
relationships could be well defined
by using the concept of color
crystallographic groups and this
concept began in early 1984. The
color crystallographic groups
possess both symmetry and anti
symmetry operators which is used for
defining the above relationships in
a well structured manner. The
crystallographic bicolor group set
belonging to each group helps in
explaining the one index-2 group or
subgroup pair for regular
crystallographic groups. In the
bicolor crystallographic group, each
element has an associated even or
odd binary parity flag. This is
calculated and arrived based on the
product of group generator parities
that produce the element.
The concept of anti symmetry plays a
vital role in color crystallographic
groups. The term anti symmetry was
coined by Heesch and Shubnikov.
Bicolor groups also referred to as
magnetic groups helps to study and
explain in detail concurrently the
arrangement of atoms also refered as
regular symmetry along with the up
or down magnetic spin vector
orientations which is referred to as
anti symmetry for magnetic atoms in
any crystal. The use of critical net
on orbifold drawing expands in more
areas and to name one in this
direction is that it is used for
explaining the complete summary of
the structure's local and global
topology if along with the critical
net on orbifold the lattice complex
information for each critical point
site is added and also the Wyckoff
site multiplicities being recorded
on the same drawing.
There are totally thirty six cubic
crystallographic space groups and
194 space groups. However, the
thirty six cubic crystallographic
space groups are unlike the 194
space groups. This is because each
of them has body diagonal 3-fold
axes that arise from their
tetrahedral and octahedral point
groups. Also, the cubic groups'
orbifolds are uncomplicated as
compared to simpler space groups
that are derivatives of cyclic and
dihedral point groups. The different
crystal shapes taken by different
crystals is because of the
prototypes associated, for instance,
the different crystal shapes in
muscovite are due to stacking
sequence shifts and not due to
different atomic structures. The
technical term used to refer the
different shapes occupied by crystal
is polytype. Sometimes, it is also
possible to get polytype occurred
when substitution causes distortion
in the shape. Generally, structural
distortion takes place in compounds
that crystallize at different
temperatures and or pressures.
Thus, the benefits of orbifolds and
critical nets on crystallographic
orbifolds are that it gives a
detailed and concise closed-space
portrait of the topology for
crystallographic groups and simple
crystal structures. In fact, a well
detailed crystallographic orbifold
atlas, if prepared, would help in
giving and projecting the complete
tabulation of the topological
properties of crystallographic
orbifolds. This would help and would
be useful to crystallographers in
various ways for determining the
attribute of crystals.
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